(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC9
/**
* Example taken from "A Term Rewriting Approach to the Automated Termination
* Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
* and converted to Java.
*/

public class PastaC9 {
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();

while (x > 0 && y > 0) {
if (Random.random() < 42) {
x--;
y = Random.random();
} else {
y--;
}
}
}
}


public class Random {
static String[] args;
static int index = 0;

public static int random() {
String string = args[index];
index++;
return string.length();
}
}


(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
PastaC9.main([Ljava/lang/String;)V: Graph of 322 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: PastaC9.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 54 rules for P and 0 rules for R.


P rules:
1005_0_main_LE(EOS(STATIC_1005), i385, i376, i385) → 1007_0_main_LE(EOS(STATIC_1007), i385, i376, i385)
1007_0_main_LE(EOS(STATIC_1007), i385, i376, i385) → 1011_0_main_Load(EOS(STATIC_1011), i385, i376) | >(i385, 0)
1011_0_main_Load(EOS(STATIC_1011), i385, i376) → 1014_0_main_LE(EOS(STATIC_1014), i385, i376, i376)
1014_0_main_LE(EOS(STATIC_1014), i385, i387, i387) → 1017_0_main_LE(EOS(STATIC_1017), i385, i387, i387)
1017_0_main_LE(EOS(STATIC_1017), i385, i387, i387) → 1022_0_main_InvokeMethod(EOS(STATIC_1022), i385, i387) | >(i387, 0)
1022_0_main_InvokeMethod(EOS(STATIC_1022), i385, i387) → 1025_0_random_FieldAccess(EOS(STATIC_1025), i385, i387)
1025_0_random_FieldAccess(EOS(STATIC_1025), i385, i387) → 1027_0_random_FieldAccess(EOS(STATIC_1027), i385, i387)
1027_0_random_FieldAccess(EOS(STATIC_1027), i385, i387) → 1030_0_random_ArrayAccess(EOS(STATIC_1030), i385, i387)
1030_0_random_ArrayAccess(EOS(STATIC_1030), i385, i387) → 1032_0_random_ArrayAccess(EOS(STATIC_1032), i385, i387)
1032_0_random_ArrayAccess(EOS(STATIC_1032), i385, i387) → 1035_0_random_Store(EOS(STATIC_1035), i385, i387, o272)
1035_0_random_Store(EOS(STATIC_1035), i385, i387, o272) → 1039_0_random_FieldAccess(EOS(STATIC_1039), i385, i387, o272)
1039_0_random_FieldAccess(EOS(STATIC_1039), i385, i387, o272) → 1042_0_random_ConstantStackPush(EOS(STATIC_1042), i385, i387, o272)
1042_0_random_ConstantStackPush(EOS(STATIC_1042), i385, i387, o272) → 1046_0_random_IntArithmetic(EOS(STATIC_1046), i385, i387, o272)
1046_0_random_IntArithmetic(EOS(STATIC_1046), i385, i387, o272) → 1050_0_random_FieldAccess(EOS(STATIC_1050), i385, i387, o272)
1050_0_random_FieldAccess(EOS(STATIC_1050), i385, i387, o272) → 1052_0_random_Load(EOS(STATIC_1052), i385, i387, o272)
1052_0_random_Load(EOS(STATIC_1052), i385, i387, o272) → 1062_0_random_InvokeMethod(EOS(STATIC_1062), i385, i387, o272)
1062_0_random_InvokeMethod(EOS(STATIC_1062), i385, i387, java.lang.Object(o290sub)) → 1065_0_random_InvokeMethod(EOS(STATIC_1065), i385, i387, java.lang.Object(o290sub))
1065_0_random_InvokeMethod(EOS(STATIC_1065), i385, i387, java.lang.Object(o290sub)) → 1069_0_length_Load(EOS(STATIC_1069), i385, i387, java.lang.Object(o290sub), java.lang.Object(o290sub))
1069_0_length_Load(EOS(STATIC_1069), i385, i387, java.lang.Object(o290sub), java.lang.Object(o290sub)) → 1077_0_length_FieldAccess(EOS(STATIC_1077), i385, i387, java.lang.Object(o290sub), java.lang.Object(o290sub))
1077_0_length_FieldAccess(EOS(STATIC_1077), i385, i387, java.lang.Object(java.lang.String(o298sub, i423)), java.lang.Object(java.lang.String(o298sub, i423))) → 1078_0_length_FieldAccess(EOS(STATIC_1078), i385, i387, java.lang.Object(java.lang.String(o298sub, i423)), java.lang.Object(java.lang.String(o298sub, i423))) | &&(>=(i423, 0), >=(i424, 0))
1078_0_length_FieldAccess(EOS(STATIC_1078), i385, i387, java.lang.Object(java.lang.String(o298sub, i423)), java.lang.Object(java.lang.String(o298sub, i423))) → 1085_0_length_Return(EOS(STATIC_1085), i385, i387, java.lang.Object(java.lang.String(o298sub, i423)), i423)
1085_0_length_Return(EOS(STATIC_1085), i385, i387, java.lang.Object(java.lang.String(o298sub, i423)), i423) → 1091_0_random_Return(EOS(STATIC_1091), i385, i387, i423)
1091_0_random_Return(EOS(STATIC_1091), i385, i387, i423) → 1093_0_main_ConstantStackPush(EOS(STATIC_1093), i385, i387, i423)
1093_0_main_ConstantStackPush(EOS(STATIC_1093), i385, i387, i423) → 1101_0_main_GE(EOS(STATIC_1101), i385, i387, i423, 42)
1101_0_main_GE(EOS(STATIC_1101), i385, i387, i439, matching1) → 1107_0_main_GE(EOS(STATIC_1107), i385, i387, i439, 42) | =(matching1, 42)
1101_0_main_GE(EOS(STATIC_1101), i385, i387, i440, matching1) → 1108_0_main_GE(EOS(STATIC_1108), i385, i387, i440, 42) | =(matching1, 42)
1107_0_main_GE(EOS(STATIC_1107), i385, i387, i439, matching1) → 1113_0_main_Inc(EOS(STATIC_1113), i385) | &&(<(i439, 42), =(matching1, 42))
1113_0_main_Inc(EOS(STATIC_1113), i385) → 1121_0_main_InvokeMethod(EOS(STATIC_1121), +(i385, -1)) | >(i385, 0)
1121_0_main_InvokeMethod(EOS(STATIC_1121), i446) → 1131_0_random_FieldAccess(EOS(STATIC_1131), i446)
1131_0_random_FieldAccess(EOS(STATIC_1131), i446) → 1451_0_random_FieldAccess(EOS(STATIC_1451), i446)
1451_0_random_FieldAccess(EOS(STATIC_1451), i446) → 1462_0_random_ArrayAccess(EOS(STATIC_1462), i446)
1462_0_random_ArrayAccess(EOS(STATIC_1462), i446) → 1470_0_random_ArrayAccess(EOS(STATIC_1470), i446)
1470_0_random_ArrayAccess(EOS(STATIC_1470), i446) → 1479_0_random_Store(EOS(STATIC_1479), i446, o482)
1479_0_random_Store(EOS(STATIC_1479), i446, o482) → 1490_0_random_FieldAccess(EOS(STATIC_1490), i446, o482)
1490_0_random_FieldAccess(EOS(STATIC_1490), i446, o482) → 1497_0_random_ConstantStackPush(EOS(STATIC_1497), i446, o482)
1497_0_random_ConstantStackPush(EOS(STATIC_1497), i446, o482) → 1508_0_random_IntArithmetic(EOS(STATIC_1508), i446, o482)
1508_0_random_IntArithmetic(EOS(STATIC_1508), i446, o482) → 1516_0_random_FieldAccess(EOS(STATIC_1516), i446, o482)
1516_0_random_FieldAccess(EOS(STATIC_1516), i446, o482) → 1523_0_random_Load(EOS(STATIC_1523), i446, o482)
1523_0_random_Load(EOS(STATIC_1523), i446, o482) → 1537_0_random_InvokeMethod(EOS(STATIC_1537), i446, o482)
1537_0_random_InvokeMethod(EOS(STATIC_1537), i446, java.lang.Object(o540sub)) → 1542_0_random_InvokeMethod(EOS(STATIC_1542), i446, java.lang.Object(o540sub))
1542_0_random_InvokeMethod(EOS(STATIC_1542), i446, java.lang.Object(o540sub)) → 1549_0_length_Load(EOS(STATIC_1549), i446, java.lang.Object(o540sub), java.lang.Object(o540sub))
1549_0_length_Load(EOS(STATIC_1549), i446, java.lang.Object(o540sub), java.lang.Object(o540sub)) → 1564_0_length_FieldAccess(EOS(STATIC_1564), i446, java.lang.Object(o540sub), java.lang.Object(o540sub))
1564_0_length_FieldAccess(EOS(STATIC_1564), i446, java.lang.Object(java.lang.String(o561sub, i726)), java.lang.Object(java.lang.String(o561sub, i726))) → 1566_0_length_FieldAccess(EOS(STATIC_1566), i446, java.lang.Object(java.lang.String(o561sub, i726)), java.lang.Object(java.lang.String(o561sub, i726))) | &&(>=(i726, 0), >=(i727, 0))
1566_0_length_FieldAccess(EOS(STATIC_1566), i446, java.lang.Object(java.lang.String(o561sub, i726)), java.lang.Object(java.lang.String(o561sub, i726))) → 1573_0_length_Return(EOS(STATIC_1573), i446, java.lang.Object(java.lang.String(o561sub, i726)), i726)
1573_0_length_Return(EOS(STATIC_1573), i446, java.lang.Object(java.lang.String(o561sub, i726)), i726) → 1577_0_random_Return(EOS(STATIC_1577), i446, i726)
1577_0_random_Return(EOS(STATIC_1577), i446, i726) → 1578_0_main_Store(EOS(STATIC_1578), i446, i726)
1578_0_main_Store(EOS(STATIC_1578), i446, i726) → 1583_0_main_JMP(EOS(STATIC_1583), i446, i726)
1583_0_main_JMP(EOS(STATIC_1583), i446, i726) → 1592_0_main_Load(EOS(STATIC_1592), i446, i726)
1592_0_main_Load(EOS(STATIC_1592), i446, i726) → 1000_0_main_Load(EOS(STATIC_1000), i446, i726)
1000_0_main_Load(EOS(STATIC_1000), i375, i376) → 1005_0_main_LE(EOS(STATIC_1005), i375, i376, i375)
1108_0_main_GE(EOS(STATIC_1108), i385, i387, i440, matching1) → 1115_0_main_Inc(EOS(STATIC_1115), i385, i387) | &&(>=(i440, 42), =(matching1, 42))
1115_0_main_Inc(EOS(STATIC_1115), i385, i387) → 1123_0_main_JMP(EOS(STATIC_1123), i385, +(i387, -1)) | >(i387, 0)
1123_0_main_JMP(EOS(STATIC_1123), i385, i447) → 1140_0_main_Load(EOS(STATIC_1140), i385, i447)
1140_0_main_Load(EOS(STATIC_1140), i385, i447) → 1000_0_main_Load(EOS(STATIC_1000), i385, i447)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.


P rules:
1005_0_main_LE(EOS(STATIC_1005), x0, x1, x0) → 1005_0_main_LE(EOS(STATIC_1005), +(x0, -1), x2, +(x0, -1)) | &&(&&(>(+(x2, 1), 0), >(x1, 0)), >(x0, 0))
1005_0_main_LE(EOS(STATIC_1005), x0, x1, x0) → 1005_0_main_LE(EOS(STATIC_1005), x0, +(x1, -1), x0) | &&(>(x1, 0), >(x0, 0))
R rules:

Filtered ground terms:



1005_0_main_LE(x1, x2, x3, x4) → 1005_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_1005_0_main_LE1(x1, x2, x3, x4, x5) → Cond_1005_0_main_LE1(x1, x3, x4, x5)
Cond_1005_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_1005_0_main_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:



1005_0_main_LE(x1, x2, x3) → 1005_0_main_LE(x2, x3)
Cond_1005_0_main_LE(x1, x2, x3, x4, x5) → Cond_1005_0_main_LE(x1, x3, x4, x5)
Cond_1005_0_main_LE1(x1, x2, x3, x4) → Cond_1005_0_main_LE1(x1, x3, x4)

Filtered unneeded arguments:



Cond_1005_0_main_LE(x1, x2, x3, x4) → Cond_1005_0_main_LE(x1, x3, x4)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.


P rules:
1005_0_main_LE(x1, x0) → 1005_0_main_LE(x2, +(x0, -1)) | &&(&&(>(x2, -1), >(x1, 0)), >(x0, 0))
1005_0_main_LE(x1, x0) → 1005_0_main_LE(+(x1, -1), x0) | &&(>(x1, 0), >(x0, 0))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.


P rules:
1005_0_MAIN_LE(x1, x0) → COND_1005_0_MAIN_LE(&&(&&(>(x2, -1), >(x1, 0)), >(x0, 0)), x1, x0, x2)
COND_1005_0_MAIN_LE(TRUE, x1, x0, x2) → 1005_0_MAIN_LE(x2, +(x0, -1))
1005_0_MAIN_LE(x1, x0) → COND_1005_0_MAIN_LE1(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_1005_0_MAIN_LE1(TRUE, x1, x0) → 1005_0_MAIN_LE(+(x1, -1), x0)
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(x2[0] > -1 && x1[0] > 0 && x0[0] > 0, x1[0], x0[0], x2[0])
(1): COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], x0[1] + -1)
(2): 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(x1[3] + -1, x0[3])

(0) -> (1), if (x2[0] > -1 && x1[0] > 0 && x0[0] > 0x1[0]* x1[1]x0[0]* x0[1]x2[0]* x2[1])


(1) -> (0), if (x2[1]* x1[0]x0[1] + -1* x0[0])


(1) -> (2), if (x2[1]* x1[2]x0[1] + -1* x0[2])


(2) -> (3), if (x1[2] > 0 && x0[2] > 0x1[2]* x1[3]x0[2]* x0[3])


(3) -> (0), if (x1[3] + -1* x1[0]x0[3]* x0[0])


(3) -> (2), if (x1[3] + -1* x1[2]x0[3]* x0[2])



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7571ccae Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair 1005_0_MAIN_LE(x1, x0) → COND_1005_0_MAIN_LE(&&(&&(>(x2, -1), >(x1, 0)), >(x0, 0)), x1, x0, x2) the following chains were created:
  • We consider the chain COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1)), 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0]), COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1)) which results in the following constraint:

    (1)    (x2[1]=x1[0]+(x0[1], -1)=x0[0]&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0))=TRUEx1[0]=x1[1]1x0[0]=x0[1]1x2[0]=x2[1]11005_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧1005_0_MAIN_LE(x1[0], x0[0])≥COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])∧(UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥))



    We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (>(+(x0[1], -1), 0)=TRUE>(x2[0], -1)=TRUE>(x1[0], 0)=TRUE1005_0_MAIN_LE(x1[0], +(x0[1], -1))≥NonInfC∧1005_0_MAIN_LE(x1[0], +(x0[1], -1))≥COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(+(x0[1], -1), 0)), x1[0], +(x0[1], -1), x2[0])∧(UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (x0[1] + [-2] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (x0[1] + [-2] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (x0[1] + [-2] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x0[1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[1] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (x0[1] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[1] ≥ 0∧[(-1)bso_18] ≥ 0)



  • We consider the chain COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]), 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0]), COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1)) which results in the following constraint:

    (8)    (+(x1[3], -1)=x1[0]x0[3]=x0[0]&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]x2[0]=x2[1]1005_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧1005_0_MAIN_LE(x1[0], x0[0])≥COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])∧(UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥))



    We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(x0[0], 0)=TRUE>(x2[0], -1)=TRUE>(+(x1[3], -1), 0)=TRUE1005_0_MAIN_LE(+(x1[3], -1), x0[0])≥NonInfC∧1005_0_MAIN_LE(+(x1[3], -1), x0[0])≥COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(+(x1[3], -1), 0)), >(x0[0], 0)), +(x1[3], -1), x0[0], x2[0])∧(UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (11)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (13)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_1005_0_MAIN_LE(TRUE, x1, x0, x2) → 1005_0_MAIN_LE(x2, +(x0, -1)) the following chains were created:
  • We consider the chain 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0]), COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1)), 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0]) which results in the following constraint:

    (15)    (&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]x2[0]=x2[1]x2[1]=x1[0]1+(x0[1], -1)=x0[0]1COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1])≥NonInfC∧COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1])≥1005_0_MAIN_LE(x2[1], +(x0[1], -1))∧(UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥))



    We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (16)    (>(x0[0], 0)=TRUE>(x2[0], -1)=TRUE>(x1[0], 0)=TRUECOND_1005_0_MAIN_LE(TRUE, x1[0], x0[0], x2[0])≥NonInfC∧COND_1005_0_MAIN_LE(TRUE, x1[0], x0[0], x2[0])≥1005_0_MAIN_LE(x2[0], +(x0[0], -1))∧(UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥))



    We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (17)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (18)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (19)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (20)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (21)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



  • We consider the chain 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0]), COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1)), 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (22)    (&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]x2[0]=x2[1]x2[1]=x1[2]+(x0[1], -1)=x0[2]COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1])≥NonInfC∧COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1])≥1005_0_MAIN_LE(x2[1], +(x0[1], -1))∧(UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥))



    We simplified constraint (22) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (23)    (>(x0[0], 0)=TRUE>(x2[0], -1)=TRUE>(x1[0], 0)=TRUECOND_1005_0_MAIN_LE(TRUE, x1[0], x0[0], x2[0])≥NonInfC∧COND_1005_0_MAIN_LE(TRUE, x1[0], x0[0], x2[0])≥1005_0_MAIN_LE(x2[0], +(x0[0], -1))∧(UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥))



    We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (24)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (25)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (26)    (x0[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)







For Pair 1005_0_MAIN_LE(x1, x0) → COND_1005_0_MAIN_LE1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]) which results in the following constraint:

    (29)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]1005_0_MAIN_LE(x1[2], x0[2])≥NonInfC∧1005_0_MAIN_LE(x1[2], x0[2])≥COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



    We simplified constraint (29) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (30)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE1005_0_MAIN_LE(x1[2], x0[2])≥NonInfC∧1005_0_MAIN_LE(x1[2], x0[2])≥COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



    We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (34)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)







For Pair COND_1005_0_MAIN_LE1(TRUE, x1, x0) → 1005_0_MAIN_LE(+(x1, -1), x0) the following chains were created:
  • We consider the chain 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]), 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0]) which results in the following constraint:

    (36)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]+(x1[3], -1)=x1[0]x0[3]=x0[0]COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3])≥1005_0_MAIN_LE(+(x1[3], -1), x0[3])∧(UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥))



    We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (37)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUECOND_1005_0_MAIN_LE1(TRUE, x1[2], x0[2])≥NonInfC∧COND_1005_0_MAIN_LE1(TRUE, x1[2], x0[2])≥1005_0_MAIN_LE(+(x1[2], -1), x0[2])∧(UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥))



    We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (41)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



  • We consider the chain 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]), 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (43)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]+(x1[3], -1)=x1[2]1x0[3]=x0[2]1COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3])≥1005_0_MAIN_LE(+(x1[3], -1), x0[3])∧(UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥))



    We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (44)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUECOND_1005_0_MAIN_LE1(TRUE, x1[2], x0[2])≥NonInfC∧COND_1005_0_MAIN_LE1(TRUE, x1[2], x0[2])≥1005_0_MAIN_LE(+(x1[2], -1), x0[2])∧(UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥))



    We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (45)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (46)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (47)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (48)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



    We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (49)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 1005_0_MAIN_LE(x1, x0) → COND_1005_0_MAIN_LE(&&(&&(>(x2, -1), >(x1, 0)), >(x0, 0)), x1, x0, x2)
    • (x0[1] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[1] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_1005_0_MAIN_LE(TRUE, x1, x0, x2) → 1005_0_MAIN_LE(x2, +(x0, -1))
    • (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
    • (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(x2[1], +(x0[1], -1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

  • 1005_0_MAIN_LE(x1, x0) → COND_1005_0_MAIN_LE1(&&(>(x1, 0), >(x0, 0)), x1, x0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

  • COND_1005_0_MAIN_LE1(TRUE, x1, x0) → 1005_0_MAIN_LE(+(x1, -1), x0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)




The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = [3]   
POL(1005_0_MAIN_LE(x1, x2)) = [-1] + x2   
POL(COND_1005_0_MAIN_LE(x1, x2, x3, x4)) = [-1] + x3   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(COND_1005_0_MAIN_LE1(x1, x2, x3)) = [-1] + x3   

The following pairs are in P>:

COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1))

The following pairs are in Pbound:

1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])
COND_1005_0_MAIN_LE(TRUE, x1[1], x0[1], x2[1]) → 1005_0_MAIN_LE(x2[1], +(x0[1], -1))
1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3])

The following pairs are in P:

1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(&&(&&(>(x2[0], -1), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0], x2[0])
1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3])

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1005_0_MAIN_LE(x1[0], x0[0]) → COND_1005_0_MAIN_LE(x2[0] > -1 && x1[0] > 0 && x0[0] > 0, x1[0], x0[0], x2[0])
(2): 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(x1[3] + -1, x0[3])

(3) -> (0), if (x1[3] + -1* x1[0]x0[3]* x0[0])


(3) -> (2), if (x1[3] + -1* x1[2]x0[3]* x0[2])


(2) -> (3), if (x1[2] > 0 && x0[2] > 0x1[2]* x1[3]x0[2]* x0[3])



The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(x1[3] + -1, x0[3])
(2): 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

(3) -> (2), if (x1[3] + -1* x1[2]x0[3]* x0[2])


(2) -> (3), if (x1[2] > 0 && x0[2] > 0x1[2]* x1[3]x0[2]* x0[3])



The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7571ccae Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]) the following chains were created:
  • We consider the chain 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]), 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (1)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]+(x1[3], -1)=x1[2]1x0[3]=x0[2]1COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3])≥1005_0_MAIN_LE(+(x1[3], -1), x0[3])∧(UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥))



    We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUECOND_1005_0_MAIN_LE1(TRUE, x1[2], x0[2])≥NonInfC∧COND_1005_0_MAIN_LE1(TRUE, x1[2], x0[2])≥1005_0_MAIN_LE(+(x1[2], -1), x0[2])∧(UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x0[2] + [bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]x0[2] + [bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)







For Pair 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
  • We consider the chain 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3]) which results in the following constraint:

    (8)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]1005_0_MAIN_LE(x1[2], x0[2])≥NonInfC∧1005_0_MAIN_LE(x1[2], x0[2])≥COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



    We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE1005_0_MAIN_LE(x1[2], x0[2])≥NonInfC∧1005_0_MAIN_LE(x1[2], x0[2])≥COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[2] + [bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[2] + [bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[2] + [bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (13)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x0[2] + [bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [bni_14]x0[2] + [bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1005_0_MAIN_LE(+(x1[3], -1), x0[3])), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]x0[2] + [bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)

  • 1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [bni_14]x0[2] + [bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)




The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [3]   
POL(FALSE) = [3]   
POL(COND_1005_0_MAIN_LE1(x1, x2, x3)) = [-1] + x3 + x2   
POL(1005_0_MAIN_LE(x1, x2)) = x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(&&(x1, x2)) = [2]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in Pbound:

COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3])
1005_0_MAIN_LE(x1[2], x0[2]) → COND_1005_0_MAIN_LE1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P:

COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(+(x1[3], -1), x0[3])

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1

(12) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_1005_0_MAIN_LE1(TRUE, x1[3], x0[3]) → 1005_0_MAIN_LE(x1[3] + -1, x0[3])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE